Find the volume v of the solid obtained by rotating the...

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 5x4, y = 5x, x ≥ 0; about the x-axis Find the area of the region enclosed by the given curves. y = 3 cos(πx), y = 12x2 − 3 Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. 2x = y2, x = 0, y = 5; about the y-axis Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = x3, y = 0, x = 1, x = 2

Transcript

00:02 Hi, here in this question we are given that we need to find the volume for the first one and equation of 5 is 5 equals to 5x to the power 4 and 5 equals to 5x for x greater than 0.

00:17 Now comparing this to equation we have 5 times x to the power 4 equals to 5x which gives the value of x to be 0 and 1.

00:26 So here our r1 of x equals to 5x and r2 of x is equal to 5 to 5 times x to the power 4.

00:37 So here dh would be equal to dx.

00:42 Hence here our volume v equals to integration over 0 to 1 5 times of r1 of square minus r2 of square dh.

00:54 So here in our case we have integration.

00:57 Over 0 to 1, pi as it is 25x square minus 25 x to the power 9 dx.

01:06 So here on integrating this we have pi times 25 x cube upon 3 minus 25 x to the power 9 upon 9 and the value is from 0 to 1.

01:16 Now we have volume v equals to 56 5 upon 9 cubic unit.

01:23 So this is our required first answer.

01:26 Now moving further for the part b here for the part b we are given that we need to find the area bounded by y equals to six times cos of pi x and yv equals to 12 x square minus three on comparing both this equation here we have six times cost of pi x equals to 12 x square minus three on solving this we have x equals to plus or minus 1 by 2 so here in our case our area would be area equals to integration over minus 1 by 2 to 1 by 2 and here we have 6 times cos of pi x minus 12 x cube 12 x square minus 3 and here whole term is multiplied with dx now further on integrating this we have 6 side pi of x divided by pi minus 12 x cube upon 3 plus 3x and the value is from minus 1 by 2 to 1 by 2.

02:36 Now here giving this value and simplifying we have 12 upon 5 minus 1 plus 3.

02:42 So here on simplifying this we can say that we got our required area to be 2 times 6 plus 5 divided by pi.

02:51 Now further moving towards the third part of a question which is 2x equals to y square x equals to 0 and y equals to 5 and this is to be rotated about 5x.

03:05 We need to calculate the volume using cylindrical method.

03:09 Now here we have x equals to 0...

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